The Distribution of Ascents of Size d or More in Partitions of n

A partition of a positive integer n is a finite sequence of positive integers a 1 , a 2 , . . ., a k such that a 1 + a 2 +ċ ċ ċ+ a k = n and a i +1 ≥ a i for all i . Let d be a fixed positive integer. We say that we have an ascent of size d or more if a i +1 ≥ a i + d . We determine the mean, the variance and the limiting distribution of the number of ascents of size d or more (equivalently, the number of distinct part sizes of multiplicity d or more) in the partitions of n .